Inapproximability results for combinatorial auctions with submodular utility functions

We consider the following allocation problem arising in the setting of combinatorial auctions: a set of goods is to be allocated to a set of players so as to maximize the sum of the utilities of the players (i.e., the social welfare). In the case when the utility of each player is a monotone submodular function, we prove that there is no polynomial time approximation algorithm which approximates the maximum social welfare by a factor better than 1 − 1/e ' 0.632, unless P = NP. Our result is based on a reduction from a multi-prover proof system for MAX-3-COLORING.